Generators of the product of two Schoenflies motion groups

Schoenflies motion is often termed X-motion for conciseness. A set of X-motions with a given direction of its axes of rotations has the algebraic properties of a Lie group for the composition product of rigid-body motions or displacements. The product of two X-subgroups, which is the mathematical model of a serial concatenation of two kinematic chains generating two distinct X-motions, characterizes a noteworthy type of 5-dimensional (5D) displacement set called double Schoenflies motion or X–X motion for brevity. This X–X motion set is a 5D submanifold of the displacement 6D Lie group. Such a motion type includes any spatial translation (3T) and any two sequential rotations (2R) provided that the axes of rotation are parallel to two fixed independent vectors. This motion set also contains the rotations that are products of the foregoing two rotations. In the paper, some preliminary fundamentals on the 4D X-motion are recalled; the 5D set of X–X motions is emphasized. Then implementing serial arrays of one-dof Reuleaux pairs and hinged parallelograms, we enumerate all serial mechanical generators of X–X motion, which have no redundant internal mobility. Based on the group-theoretic concepts, one can differentiate two families of irreducible representations of an X–X motion. One family is realized by twenty-one open chains including the doubly planar motion generators as special cases. The other is generally classified into eight major categories in which one hundred and six distinct open chains generating X–X motion are revealed and nineteen more ones having at least one parallelogram are derived from them. Meanwhile, these kinematic chains are graphically displayed for a possible use in the structural synthesis of parallel manipulators.     

锥体积泛函与仿射不等式(英文)

本文研究了凸多胞形的锥体积泛函.利用投影体以及Lutwak、杨和张最近所建立的仿射等周不等式,得到了刻划平行四边形特征的一个崭新不等式和用锥体积泛函以及投影体的体积所表达的关于配极体体积的严格下界.     

An inequality for distances between 2n points and the Aleksandrov-Rassias problem

In this paper, we introduce an inequality for distances between every two points among the given 2n points and apply this inequality to a partial solution of the Aleksandrov-Rassias problem, which was first posed by Th.M. Rassias.     

The meaning of computer search in the study of some classes of IM-quasigroups

The IM-quasigroup C(q) for or is a GS-quasigroup. Some interesting geometric concepts can be introduced in a general GS-quasigroup and their nice geometric representations can be given in the mentioned GS-quasigroup which justifies the research of this quasigroup. The concept of a parallelogram can be defined by means of several equivalent formulae. Some of them can be obtained by means of a computer. We shall choose a suitable formula for the definition of a parallelogram which allows the characterization of GS-quasigroups by means of commutative groups.      

Generalized Gaussian quadrature rules over two-dimensional regions with linear sides

This paper presents a generalized Gaussian quadrature method for numerical integration over triangular, parallelogram and quadrilateral elements with linear sides. In order to derive the quadrature rule, a general transformation of the regions, R₁ = {(x, y)∣a ? x ? b, g(x) ? y ? h(x)} and R₂ = {(x, y)∣a ? y ? b, g(y) ? x ? h(y)}, where g(x), h(x), g(y) and h(y) are linear functions, is given from (x, y) space to a square in (ξ, η) space, S: {(ξ, η)∣0 ? ξ ? 1, 0 ? η ? 1}. Generlized Gaussian quadrature nodes and weights introduced by Ma et.al. in 1997 are used in the product formula presented in this paper to evaluate the integral over S, as it is proved to give more accurate results than the classical Gauss Legendre nodes and weights. The method can be used to integrate a wide class of functions including smooth functions and functions with end-point singularities, over any two-dimensional region, bounded by linear sides. The performance of the method is illustrated for different functions over different two-dimensional regions with numerical examples.     

Combinatorics of non-ambiguous trees

This article investigates combinatorial properties of non-ambiguous trees. These objects we define may be seen either as binary trees drawn on a grid with some constraints, or as a subset of the tree-like tableaux previously defined by Aval, Boussicault and Nadeau. The enumeration of non-ambiguous trees satisfying some additional constraints allows us to give elegant combinatorial proofs of identities due to Carlitz, and to Ehrenborg and Steingrímsson. We also provide a hook formula to count the number of non-ambiguous trees with a given underlying tree. Finally, we use non-ambiguous trees to describe a very natural bijection between parallelogram polyominoes and binary trees.